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Theorem filconn 23607
Description: A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
filconn (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

Proof of Theorem filconn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2 filunibas 23605 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
32fveq2d 6894 . . 3 (𝐹 ∈ (Fil‘𝑋) → (Fil‘ 𝐹) = (Fil‘𝑋))
41, 3eleqtrrd 2834 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘ 𝐹))
5 nss 4045 . . . . . . . 8 𝑥 ⊆ {∅} ↔ ∃𝑦(𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅}))
6 simpll 763 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝐹 ∈ (Fil‘ 𝐹))
7 ssel2 3976 . . . . . . . . . . . . . . . . 17 ((𝑥 ⊆ (𝐹 ∪ {∅}) ∧ 𝑦𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
87adantll 710 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
9 elun 4147 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐹 ∪ {∅}) ↔ (𝑦𝐹𝑦 ∈ {∅}))
108, 9sylib 217 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → (𝑦𝐹𝑦 ∈ {∅}))
1110orcomd 867 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → (𝑦 ∈ {∅} ∨ 𝑦𝐹))
1211ord 860 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦𝑥) → (¬ 𝑦 ∈ {∅} → 𝑦𝐹))
1312impr 453 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦𝐹)
14 uniss 4915 . . . . . . . . . . . . . 14 (𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 (𝐹 ∪ {∅}))
1514ad2antlr 723 . . . . . . . . . . . . 13 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥 (𝐹 ∪ {∅}))
16 uniun 4933 . . . . . . . . . . . . . 14 (𝐹 ∪ {∅}) = ( 𝐹 {∅})
17 0ex 5306 . . . . . . . . . . . . . . . 16 ∅ ∈ V
1817unisn 4929 . . . . . . . . . . . . . . 15 {∅} = ∅
1918uneq2i 4159 . . . . . . . . . . . . . 14 ( 𝐹 {∅}) = ( 𝐹 ∪ ∅)
20 un0 4389 . . . . . . . . . . . . . 14 ( 𝐹 ∪ ∅) = 𝐹
2116, 19, 203eqtrri 2763 . . . . . . . . . . . . 13 𝐹 = (𝐹 ∪ {∅})
2215, 21sseqtrrdi 4032 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥 𝐹)
23 elssuni 4940 . . . . . . . . . . . . 13 (𝑦𝑥𝑦 𝑥)
2423ad2antrl 724 . . . . . . . . . . . 12 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦 𝑥)
25 filss 23577 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘ 𝐹) ∧ (𝑦𝐹 𝑥 𝐹𝑦 𝑥)) → 𝑥𝐹)
266, 13, 22, 24, 25syl13anc 1370 . . . . . . . . . . 11 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥𝐹)
27 elun1 4175 . . . . . . . . . . 11 ( 𝑥𝐹 𝑥 ∈ (𝐹 ∪ {∅}))
2826, 27syl 17 . . . . . . . . . 10 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑥 ∈ (𝐹 ∪ {∅}))
2928ex 411 . . . . . . . . 9 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → ((𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
3029exlimdv 1934 . . . . . . . 8 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (∃𝑦(𝑦𝑥 ∧ ¬ 𝑦 ∈ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
315, 30biimtrid 241 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (¬ 𝑥 ⊆ {∅} → 𝑥 ∈ (𝐹 ∪ {∅})))
32 uni0b 4936 . . . . . . . 8 ( 𝑥 = ∅ ↔ 𝑥 ⊆ {∅})
33 ssun2 4172 . . . . . . . . . 10 {∅} ⊆ (𝐹 ∪ {∅})
3417snid 4663 . . . . . . . . . 10 ∅ ∈ {∅}
3533, 34sselii 3978 . . . . . . . . 9 ∅ ∈ (𝐹 ∪ {∅})
36 eleq1 2819 . . . . . . . . 9 ( 𝑥 = ∅ → ( 𝑥 ∈ (𝐹 ∪ {∅}) ↔ ∅ ∈ (𝐹 ∪ {∅})))
3735, 36mpbiri 257 . . . . . . . 8 ( 𝑥 = ∅ → 𝑥 ∈ (𝐹 ∪ {∅}))
3832, 37sylbir 234 . . . . . . 7 (𝑥 ⊆ {∅} → 𝑥 ∈ (𝐹 ∪ {∅}))
3931, 38pm2.61d2 181 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → 𝑥 ∈ (𝐹 ∪ {∅}))
4039ex 411 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
4140alrimiv 1928 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → ∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})))
42 filin 23578 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
43 elun1 4175 . . . . . . . . . 10 ((𝑥𝑦) ∈ 𝐹 → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
4442, 43syl 17 . . . . . . . . 9 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
45443expa 1116 . . . . . . . 8 (((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) ∧ 𝑦𝐹) → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
4645ralrimiva 3144 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → ∀𝑦𝐹 (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
47 elsni 4644 . . . . . . . . 9 (𝑦 ∈ {∅} → 𝑦 = ∅)
48 ineq2 4205 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑥𝑦) = (𝑥 ∩ ∅))
49 in0 4390 . . . . . . . . . . 11 (𝑥 ∩ ∅) = ∅
5048, 49eqtrdi 2786 . . . . . . . . . 10 (𝑦 = ∅ → (𝑥𝑦) = ∅)
5150, 35eqeltrdi 2839 . . . . . . . . 9 (𝑦 = ∅ → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5247, 51syl 17 . . . . . . . 8 (𝑦 ∈ {∅} → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5352rgen 3061 . . . . . . 7 𝑦 ∈ {∅} (𝑥𝑦) ∈ (𝐹 ∪ {∅})
54 ralun 4191 . . . . . . 7 ((∀𝑦𝐹 (𝑥𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑦 ∈ {∅} (𝑥𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5546, 53, 54sylancl 584 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
5655ralrimiva 3144 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → ∀𝑥𝐹𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
57 elsni 4644 . . . . . . 7 (𝑥 ∈ {∅} → 𝑥 = ∅)
58 ineq1 4204 . . . . . . . . . 10 (𝑥 = ∅ → (𝑥𝑦) = (∅ ∩ 𝑦))
59 0in 4392 . . . . . . . . . 10 (∅ ∩ 𝑦) = ∅
6058, 59eqtrdi 2786 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝑦) = ∅)
6160, 35eqeltrdi 2839 . . . . . . . 8 (𝑥 = ∅ → (𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6261ralrimivw 3148 . . . . . . 7 (𝑥 = ∅ → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6357, 62syl 17 . . . . . 6 (𝑥 ∈ {∅} → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6463rgen 3061 . . . . 5 𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅})
65 ralun 4191 . . . . 5 ((∀𝑥𝐹𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
6656, 64, 65sylancl 584 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))
67 p0ex 5381 . . . . . 6 {∅} ∈ V
68 unexg 7738 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ {∅} ∈ V) → (𝐹 ∪ {∅}) ∈ V)
6967, 68mpan2 687 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → (𝐹 ∪ {∅}) ∈ V)
70 istopg 22617 . . . . 5 ((𝐹 ∪ {∅}) ∈ V → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))))
7169, 70syl 17 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥𝑦) ∈ (𝐹 ∪ {∅}))))
7241, 66, 71mpbir2and 709 . . 3 (𝐹 ∈ (Fil‘ 𝐹) → (𝐹 ∪ {∅}) ∈ Top)
7321cldopn 22755 . . . . . . . 8 (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → ( 𝐹𝑥) ∈ (𝐹 ∪ {∅}))
74 elun 4147 . . . . . . . 8 (( 𝐹𝑥) ∈ (𝐹 ∪ {∅}) ↔ (( 𝐹𝑥) ∈ 𝐹 ∨ ( 𝐹𝑥) ∈ {∅}))
7573, 74sylib 217 . . . . . . 7 (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (( 𝐹𝑥) ∈ 𝐹 ∨ ( 𝐹𝑥) ∈ {∅}))
76 elun 4147 . . . . . . . . . 10 (𝑥 ∈ (𝐹 ∪ {∅}) ↔ (𝑥𝐹𝑥 ∈ {∅}))
77 filfbas 23572 . . . . . . . . . . . . . 14 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ∈ (fBas‘ 𝐹))
78 fbncp 23563 . . . . . . . . . . . . . 14 ((𝐹 ∈ (fBas‘ 𝐹) ∧ 𝑥𝐹) → ¬ ( 𝐹𝑥) ∈ 𝐹)
7977, 78sylan 578 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → ¬ ( 𝐹𝑥) ∈ 𝐹)
8079pm2.21d 121 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥𝐹) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅))
8180ex 411 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥𝐹 → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8257a1i13 27 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ∈ {∅} → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8381, 82jaod 855 . . . . . . . . . 10 (𝐹 ∈ (Fil‘ 𝐹) → ((𝑥𝐹𝑥 ∈ {∅}) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8476, 83biimtrid 241 . . . . . . . . 9 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ∈ (𝐹 ∪ {∅}) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅)))
8584imp 405 . . . . . . . 8 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (( 𝐹𝑥) ∈ 𝐹𝑥 = ∅))
86 elsni 4644 . . . . . . . . 9 (( 𝐹𝑥) ∈ {∅} → ( 𝐹𝑥) = ∅)
87 elssuni 4940 . . . . . . . . . . . 12 (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 (𝐹 ∪ {∅}))
8887, 21sseqtrrdi 4032 . . . . . . . . . . 11 (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 𝐹)
8988adantl 480 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → 𝑥 𝐹)
90 ssdif0 4362 . . . . . . . . . . 11 ( 𝐹𝑥 ↔ ( 𝐹𝑥) = ∅)
9190biimpri 227 . . . . . . . . . 10 (( 𝐹𝑥) = ∅ → 𝐹𝑥)
92 eqss 3996 . . . . . . . . . . 11 (𝑥 = 𝐹 ↔ (𝑥 𝐹 𝐹𝑥))
9392simplbi2 499 . . . . . . . . . 10 (𝑥 𝐹 → ( 𝐹𝑥𝑥 = 𝐹))
9489, 91, 93syl2im 40 . . . . . . . . 9 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (( 𝐹𝑥) = ∅ → 𝑥 = 𝐹))
9586, 94syl5 34 . . . . . . . 8 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (( 𝐹𝑥) ∈ {∅} → 𝑥 = 𝐹))
9685, 95orim12d 961 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((( 𝐹𝑥) ∈ 𝐹 ∨ ( 𝐹𝑥) ∈ {∅}) → (𝑥 = ∅ ∨ 𝑥 = 𝐹)))
9775, 96syl5 34 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (𝑥 = ∅ ∨ 𝑥 = 𝐹)))
9897expimpd 452 . . . . 5 (𝐹 ∈ (Fil‘ 𝐹) → ((𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))) → (𝑥 = ∅ ∨ 𝑥 = 𝐹)))
99 elin 3963 . . . . 5 (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ↔ (𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))))
100 vex 3476 . . . . . 6 𝑥 ∈ V
101100elpr 4650 . . . . 5 (𝑥 ∈ {∅, 𝐹} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐹))
10298, 99, 1013imtr4g 295 . . . 4 (𝐹 ∈ (Fil‘ 𝐹) → (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) → 𝑥 ∈ {∅, 𝐹}))
103102ssrdv 3987 . . 3 (𝐹 ∈ (Fil‘ 𝐹) → ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, 𝐹})
10421isconn2 23138 . . 3 ((𝐹 ∪ {∅}) ∈ Conn ↔ ((𝐹 ∪ {∅}) ∈ Top ∧ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, 𝐹}))
10572, 103, 104sylanbrc 581 . 2 (𝐹 ∈ (Fil‘ 𝐹) → (𝐹 ∪ {∅}) ∈ Conn)
1064, 105syl 17 1 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 843  w3a 1085  wal 1537   = wceq 1539  wex 1779  wcel 2104  wral 3059  Vcvv 3472  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321  {csn 4627  {cpr 4629   cuni 4907  cfv 6542  fBascfbas 21132  Topctop 22615  Clsdccld 22740  Conncconn 23135  Filcfil 23569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-fbas 21141  df-top 22616  df-cld 22743  df-conn 23136  df-fil 23570
This theorem is referenced by:  ufildr  23655
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